Computer methods and Borel summability applied to Feigenbaum’s equation.

*(English)*Zbl 0598.58040
Lecture Notes in Physics, 227. Berlin etc.: Springer-Verlag. xiv, 297 p. DM 36.50 (1985).

This book considers the Feigenbaum universality for functions of \(| x|^{2N}\) with \(N\) very large, i.e. the functional equation

\[ f_N(z)=\frac{1}{\lambda_N} f_N \left(\left[f_N(| \lambda_N|^{2N}z)\right]^{2N}\right),\quad f_N(0)=1. \] For large \(N\) this problem is viewed as a perturbation of the case \(N=\infty\), which turns out to be singular. The main parts of the book are: 1. Feigenbaum’s universality, 2. Ecalle’s theory of resurgent functions, 3. constructive aspects of Borel summation, 4. techniques for computer-assisted proofs.

Of particular interest is the chapter dealing with computer-assisted proofs. Here the intention is to prove a theorem by invoking the contraction mapping principle in Banach space. The computer is used to give rigorous bounds for \(\| f_0-Kf_0\|\) and \(\| DK_f\|\), where \(K\) is the operator defining the equation of interest \(Kf=f\) and \(DK_f\) the tangent map of \(K\) at \(f\). The book includes many FORTRAN subroutines and programs, which may be helpful also for other problems. For the readers it might be useful to have the programs available in machine-readable form.

\[ f_N(z)=\frac{1}{\lambda_N} f_N \left(\left[f_N(| \lambda_N|^{2N}z)\right]^{2N}\right),\quad f_N(0)=1. \] For large \(N\) this problem is viewed as a perturbation of the case \(N=\infty\), which turns out to be singular. The main parts of the book are: 1. Feigenbaum’s universality, 2. Ecalle’s theory of resurgent functions, 3. constructive aspects of Borel summation, 4. techniques for computer-assisted proofs.

Of particular interest is the chapter dealing with computer-assisted proofs. Here the intention is to prove a theorem by invoking the contraction mapping principle in Banach space. The computer is used to give rigorous bounds for \(\| f_0-Kf_0\|\) and \(\| DK_f\|\), where \(K\) is the operator defining the equation of interest \(Kf=f\) and \(DK_f\) the tangent map of \(K\) at \(f\). The book includes many FORTRAN subroutines and programs, which may be helpful also for other problems. For the readers it might be useful to have the programs available in machine-readable form.

Reviewer: Helmut Weber (Wiesbaden)

##### MSC:

37-02 | Research exposition (monographs, survey articles) pertaining to dynamical systems and ergodic theory |

37-04 | Software, source code, etc. for problems pertaining to dynamical systems and ergodic theory |

40-04 | Software, source code, etc. for problems pertaining to sequences, series, summability |

37E20 | Universality and renormalization of dynamical systems |

40G10 | Abel, Borel and power series methods |

30E99 | Miscellaneous topics of analysis in the complex plane |